McCullum, under pressure after a dearth of recent five-day runs and yet to win a Test as skipper, bludgeoned an unbeaten 109 from just 121 balls as the hosts ran roughshod over a sub-standard West Indies attack after being asked to bat at the University Oval.
Taylor was unbeaten on 103 at the close as the pair put on 182 for the fourth wicket, seven short of the record against the West Indies.
McCullum raised his bat after bringing up his seventh Test century before being embraced mid-pitch by Taylor, who he replaced as captain.
His ton came from 101 balls with 13 boundaries and three sixes.
Taylor, who was on 36 when McCullum strode to the wicket, survived a referral on 95 to reach his ninth Test ton nine balls after McCullum.
His effort from 150 balls was his first century since McCullum replaced him at the helm and the new captain also hugged the 29-year-old after the milestone.
Taylor was happy for the under-pressure McCullum.
"It's been well documented that Brendon's been under the pump and it's nice for him to score a hundred after he has been in a lean patch and it's nice for him to do it in front of family and friends.
He played down the rivalry between the pair.
"We want to do well for the team and for the country.
"There's always going to be that comparison with us until he retires or I retire."
McCullum and Taylor both had openers Hamish Rutherford and Peter Fulton to thank.
Rutherford, batting on the ground where he made 171 on debut, raced to 62 but fell just before lunch after putting on 95 with Fulton.
It took a stunning one-handed catch from a backpedalling Narsingh Deonarine to end Rutherford's 83-ball stay which featured 10 fours and a six.
New Zealand went to lunch on 1-100 and the recalled Aaron Redmond, playing his first Test in five years, reached 20 before edging a Tino Best delivery to gully.
Fulton had made 61 when he nicked an innocuous Darren Sammy delivery to second slip.
With the Black Caps 3-194 at tea and 35 overs left, the scene was set for some Taylor and McCullum fireworks and they did not disappoint.
The day was a horror show for the West Indies who had hoped to make the most of a green-tinged pitch after winning the toss.
Best ran in hard and almost had Taylor cheaply while spinner Shane Shillingford, whose action is under scrutiny by the ICC, was tidy without being threatening.